I think I found a solution to my problem in the case of two chambers connected by a valve.

I can calculate the amplitude of the oscillation that occurres, and have given a condition that leads to the oscillation damping, and my results agree with what I see in Simulink.
If anyone would go over my mathematical formulation given in the attachments, I would greatly appreciate that.

However, I'm having trouble expanding my math to more chambers = higher dimensional cases.
So far, I have been able to calculate the amplitude of oscillation for 3 chambers, but couldn't give a precise formulation of a solution such as in the 1D case. I will upload a document of my calculations for this as soon as I type it in.

I presented some ideas in the 1D case which could be used to expand to higher dimensions. If anyone could give me ideas on how to do that, and if the way I'm trying seems to be correct or not, I would also be very thankful.